Torkel Franzen wrote:
>> Vladimir Sazonov says:
>> >It is
> >greatest mistake to identify the formal statement Con(PA)
> >with "Peano Arithmetic is consistent". But things like this
> >are permanently happening here in FOM!
>> Not only in FOM, but in mathematics. As you know, mathematicians in
> general neither know nor care about the distinctions you emphasize,
> but speak freely about the truth or falsity of the Riemann hypothesis,
> the properties of algorithms, and so on, as do people in computer
> science and other formal fields,
Yes, I explained this in terms of their illusions.
There is (practically) nothing dangerous in this way
of thinking of working mathematicians because all they
do is under a strongest control of formal systems in
which they are working.
It absolutely does not matter whether they realize this fact
or not! (However, mathematical logicians should realize this
professionally.) They DO THIS AUTOMATICALLY, (with the help
of some mechanisms of their brains, I think, and by making
notes on a paper) as YOU DO when turn bicycle pedals without
even thinking about this. It is a very good idea to be able
to do something automatically! Mathematics allows to think
mechanically, at least to check correctness of reasoning.
Instead of turning bicycle pedals we turn our thoughts.
This is the main idea of mathematics, I think.
> without reference to any formal
> theories.
See above. Especially computer scientists are so closely
related with *explicit* formalisms (like programming languages),
even more than mathematicians or logicians.
> I would say that we have no grounds for thinking it is
> possible to uphold in practice the distinction between mathematics and
> "philosophy" that you urge.
If we will not even try (as I, believe, a GOOD philosophy of
mathematics should do), we will always mix "black" with "white"
and infinitely twaddle.
A good rule: do not use the word TRUTH (or GOD) without a
very serious need or reason. I believe, there should be a
good corresponding dictum in English. (Ne pominaj Boga v sue
- in Russian - kazhetsa tak).
>> >It is completely unclear which way we could conclude (without
> >making all the necessary distinctions!) that from some
> >philosophical point of view there are arithmetical truths
> >(IN WHICH SENSE, PLEASE?) which are unprovable (IN WHICH SENSE,
> >PLEASE?).
>> Why, in the ordinary mathematical sense, of course.
Which one? TRUTH = ILLUSION or TRUTH = PROVABILITY?
PROVABLE BY HUMAN BEING or in the sense of
\exists x Proof(x,y) with Goedel formalization of
Proof predicate?
Be careful and precise, please!
When we speak of
> e.g. the possibility that Goldbach's conjecture is true but unprovable
> in PA, we are speaking of the possibility that even if every even
> number greater than two is the sum of two primes, there may be no
> formal deduction in PA of the canonical formalization of this
> statement.
I understand this phrase ONLY as a mathematical sentence
(i.e. FORMAL one, using the formal quantifier symbols,
the formally defined predicate Proof and the entailment
notion |=, etc.; see the above comments on formal character
of mathematics). Then there is nothing philosophical to
discuss. Just prove or disprove the corresponding formal
sentence in a formalism where it was formalized/formulated
(say, in ZFC).
But returning again to yours
> Why, in the ordinary mathematical sense, of course.
I recall that this was said concerning
> >from some
> >philosophical point of view ...
I have no idea how mathematically describe that
"philosophical point of view" (see above). That view
was (quasi)philosophical and therefore should be grounded
philosophically. By a GOOD PHILOSOPHY, PLEASE, making
all the necessary distinctions and having nothing to do
with (theological) beliefs (in truth of axioms or in anything
else). It should be a philosophy of SCIENCE.
What about the following principle:
Our illusions, even mathematical, cannot be formalized
in principle. (Who doubts?)
It seems Goedel's incompleteness theorems may play a role of
a rigorous mathematical counterpart (not a proof or formalization)
of this self-evident principle.
Vladimir Sazonov